dchisqApprox {DPQ}

Approximations of the (Noncentral) Chi-Squared Density

Description

Compute the density function f(x, *) of the (noncentral) chi-squared distribution.

Usage

dnchisqR     (x, df, ncp, log = FALSE,
              eps = 5e-15, termSml = 1e-10, ncpLarge = 1000)
dnchisqBessel(x, df, ncp, log = FALSE)
dchisqAsym   (x, df, ncp, log = FALSE)
dnoncentchisq(x, df, ncp, kmax = floor(ncp/2 + 5 * (ncp/2)^0.5))

Arguments

x

non-negative numeric vector.

df

degrees of freedom (parameter), a positive number.

ncp

non-centrality parameter \delta; ....

log

logical indicating if the result is desired on the log scale.

eps

positive convergence tolerance for the series expansion: Terms are added while term * q > (1-q)*eps, where q is the term's multiplication factor.

termSml

positive tolerance: in the series expansion, terms are added to the sum as long as they are not smaller than termSml * sum even when convergence according to eps had occured. This was not part of the original C code, but was added later for safeguarding against infinite loops, from PR#14105, e.g., for dchisq(2000, 2, 1000).

ncpLarge

in the case where mid underflows to 0, when log is true, or ncp >= ncpLarge, use a central approximation. In theory, an optimal choice of ncpLarge would not be arbitrarily set at 1000 (hardwired in R's dchisq() here), but possibly also depend on x or df.

kmax

the number of terms in the sum for dnoncentchisq().

Details

dnchisqR() is a pure R implementation of R's own C implementation in the sources, ‘R/src/nmath/dnchisq.c’, additionally exposing the three “tuning parameters” eps, termSml, and ncpLarge.

dnchisqBessel() implements Fisher(1928)'s exact closed form formula based on the Bessel function I_{nu}, i.e., R's besselI() function; specifically formula (29.4) in Johnson et al. (1995).

dchisqAsym() is the simple asymptotic approximation from Abramowitz and Stegun's formula 26.4.27, p. 942.

dnoncentchisq() uses the (typically defining) infinite series expansion directly, with truncation at kmax, and terms t_k which are products of a Poisson probability and a central chi-square density, i.e., terms t.k := dpois(k, lambda = ncp/2) * dchisq(x, df = 2*k + df) for k = 0, 1, ..., kmax.

Value

numeric vector similar to x, containing the (logged if log=TRUE) values of the density f(x,*).

Note

These functions are mostly of historical interest, notably as R's dchisq() was not always very accurate in the noncentral case, i.e., for ncp > 0.

Note

R's dchisq() is typically more uniformly accurate than the approximations nowadays, apart from dnchisqR() which should behave the same. There may occasionally exist small differences between dnchisqR(x, *) and dchisq(x, *) for the same parameters.

Author(s)

Martin Maechler, April 2008

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.

Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley; chapter 29, Section 3 Distribution, (29.4), p. 436.

See Also

R's own dchisq().

Examples

x <- sort(outer(c(1,2,5), 2^(-4:5)))
fRR <- dchisq  (x, 10, 2)
f.R <- dnchisqR(x, 10, 2)
all.equal(fRR, f.R, tol = 0) # 64bit Lnx (F 30): 1.723897e-16
stopifnot(all.equal(fRR, f.R, tol = 4e-15))